Given a space \(X\) and a path-connected subspace \(A\) containing the basepoint \(x_0\), show that the map \(\pi_1(A, x_0) \to \pi_1(X, x_0)\) induced by the inclusion \(A \hookrightarrow X\) is surjective iff every path in \(X\) with endpoints in \(A\) is homotopic to a path in \(A\).
Solution
All homotopies of paths below are relative to their endpoints.
1. Suppose the inclusion-induced map\[ i_*:\pi_1(A,x_0)\to\pi_1(X,x_0) \]is surjective.Let \(\gamma\) be a path in \(X\) from \(x\in A\) to \(y\in A\).
This fixes an arbitrary path of the type appearing in the desired path condition.
2. Choose paths \(a\) in \(A\) from \(x_0\) to \(x\) and \(b\) in \(A\) from \(y\) to \(x_0\).Then\[ \ell=a\cdot\gamma\cdot b \]is a loop in \(X\) based at \(x_0\).
The paths \(a\) and \(b\) exist because \(A\) is path connected. Their endpoints match those of \(\gamma\), so the concatenation is defined and begins and ends at \(x_0\).
3. There is a loop \(c\) in \(A\) based at \(x_0\) such that\[ [c]=[\ell]\in\pi_1(X,x_0). \]
This is precisely surjectivity of \(i_*\) applied to the class \([\ell]\).
4. The path \(\gamma\) is homotopic relative to endpoints in \(X\) to\[ \bar a\cdot c\cdot\bar b, \]which lies entirely in \(A\).
The equality in 3 means \[ a\cdot\gamma\cdot b\simeq c \] relative to the basepoint. Concatenate this homotopy on the left by \(\bar a\) and on the right by \(\bar b\) to obtain \[ \bar a\cdot a\cdot\gamma\cdot b\cdot\bar b \simeq \bar a\cdot c\cdot\bar b. \] The standard cancellation homotopies \[ \bar a\cdot a\simeq c_x, \qquad b\cdot\bar b\simeq c_y \] reduce the left-hand side to \(\gamma\) relative to its endpoints. Every factor on the right lies in \(A\), so the terminal path lies in \(A\).
5. Thus surjectivity of \(i_*\) implies the stated path condition.
The path \(\gamma\) in 1 was arbitrary.
6. Conversely, suppose every path in \(X\) whose endpoints lie in \(A\) is homotopic relative to endpoints to a path in \(A\).Then \(i_*\) is surjective.
Let \[ [\gamma]\in\pi_1(X,x_0) \] be arbitrary. The loop \(\gamma\) has both endpoints equal to \(x_0\in A\), so by hypothesis it is homotopic relative to endpoints to a path \(c\) in \(A\). Since the endpoints are fixed during the homotopy, \(c\) is a loop in \(A\) based at \(x_0\). Therefore \[ i_*([c])=[\gamma]. \] Every element of \(\pi_1(X,x_0)\) is thus in the image of \(i_*\).
7. Hence the two conditions are equivalent.
The forward implication is <1>1–<1>5, and the reverse implication is 6.